SciPy's Fisher's Exact P-Value Differs from Exact Formula

Question

I recently computed Fisher’s Exact Test for a 2×2 contingency table using SciPy’s built in fisher_exact() function. I’m using their example code from the SciPy docs:

>>> from scipy.stats import fisher_exact
>>> import numpy as np
>>> table = np.array([[6, 2], [1, 4]])
>>> res = fisher_exact(table, alternative='two-sided')
>>> res[1]
0.10256410256410257

Then I use the formula for Fisher’s Exact test using factorials of each pairwise sum divided by factorials of each cell count + factorial of all the counts summed. Here is a website with that formula: https://www.statology.org/fishers-exact-test/
Here is my code implementing the formula:

>>> import numpy as np
>>> from scipy.stats import fisher_exact
>>> from scipy.special import factorial
>>> table = np.array([[6, 2], [1, 4]])
>>> res = fisher_exact(table)

>>> fishers_plug_in = (factorial(table[0][0]+table[0][1])*factorial(table[0][0]+table[1][0])*factorial(table[1][0]+table[1][1])*
                   factorial(table[0][1]+table[1][1])/(factorial(table[0][0])*factorial(table[0][1])*
                   factorial(table[1][0])*factorial(table[1][1])*factorial(table[0][0]+table[0][1]+table[1][0]+table[1][1])))
>>> print(fishers_plug_in)
0.08158508158508158

Does anyone have any idea why the calculated P-values are different? My best guess is that SciPy uses some sort of approximation for the factorials either in the fishers_exact() function for larger contingency tables, but I can’t find any documentation about this.

Asked By: OmniWheel

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Answer 1

As commenter Severin Pappadeux has pointed out, the formula I used is only for the one sided test.

Here is the source for the factorial formula I used: https://www.statology.org/fishers-exact-test/
Underneath it says "The one-tailed p value for Fisher’s Exact Test". The factorial formula I used gives the probability that another random table maintaining row/column totals would be more strongly correlated than ours.

Here is the documentation for SciPy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fisher_exact.html

As you can see, by default SciPy uses the two-tailed hypothesis, which is the likelihood that data is more OR less correlated.

These are different hypothesis, and have different p-scores. It is not surprising that they are different.

Answered By: OmniWheel

SciPy's Fisher's Exact P-Value Differs from Exact Formula

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